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Theorem ofrn2 29743
Description: The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Hypotheses
Ref Expression
ofrn.1 (𝜑𝐹:𝐴𝐵)
ofrn.2 (𝜑𝐺:𝐴𝐵)
ofrn.3 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
ofrn.4 (𝜑𝐴𝑉)
Assertion
Ref Expression
ofrn2 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Proof of Theorem ofrn2
Dummy variables 𝑥 𝑦 𝑧 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ofrn.1 . . . . . . . 8 (𝜑𝐹:𝐴𝐵)
2 ffn 6198 . . . . . . . 8 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
31, 2syl 17 . . . . . . 7 (𝜑𝐹 Fn 𝐴)
43adantr 472 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝐹 Fn 𝐴)
5 simprl 811 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑎𝐴)
6 fnfvelrn 6511 . . . . . 6 ((𝐹 Fn 𝐴𝑎𝐴) → (𝐹𝑎) ∈ ran 𝐹)
74, 5, 6syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐹𝑎) ∈ ran 𝐹)
8 ofrn.2 . . . . . . . 8 (𝜑𝐺:𝐴𝐵)
9 ffn 6198 . . . . . . . 8 (𝐺:𝐴𝐵𝐺 Fn 𝐴)
108, 9syl 17 . . . . . . 7 (𝜑𝐺 Fn 𝐴)
1110adantr 472 . . . . . 6 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝐺 Fn 𝐴)
12 fnfvelrn 6511 . . . . . 6 ((𝐺 Fn 𝐴𝑎𝐴) → (𝐺𝑎) ∈ ran 𝐺)
1311, 5, 12syl2anc 696 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → (𝐺𝑎) ∈ ran 𝐺)
14 simprr 813 . . . . 5 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → 𝑧 = ((𝐹𝑎) + (𝐺𝑎)))
15 rspceov 6847 . . . . 5 (((𝐹𝑎) ∈ ran 𝐹 ∧ (𝐺𝑎) ∈ ran 𝐺𝑧 = ((𝐹𝑎) + (𝐺𝑎))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
167, 13, 14, 15syl3anc 1473 . . . 4 ((𝜑 ∧ (𝑎𝐴𝑧 = ((𝐹𝑎) + (𝐺𝑎)))) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦))
1716rexlimdvaa 3162 . . 3 (𝜑 → (∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎)) → ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
1817ss2abdv 3808 . 2 (𝜑 → {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))} ⊆ {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
19 ofrn.4 . . . . 5 (𝜑𝐴𝑉)
20 inidm 3957 . . . . 5 (𝐴𝐴) = 𝐴
21 eqidd 2753 . . . . 5 ((𝜑𝑎𝐴) → (𝐹𝑎) = (𝐹𝑎))
22 eqidd 2753 . . . . 5 ((𝜑𝑎𝐴) → (𝐺𝑎) = (𝐺𝑎))
233, 10, 19, 19, 20, 21, 22offval 7061 . . . 4 (𝜑 → (𝐹𝑓 + 𝐺) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
2423rneqd 5500 . . 3 (𝜑 → ran (𝐹𝑓 + 𝐺) = ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))))
25 eqid 2752 . . . 4 (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎)))
2625rnmpt 5518 . . 3 ran (𝑎𝐴 ↦ ((𝐹𝑎) + (𝐺𝑎))) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))}
2724, 26syl6eq 2802 . 2 (𝜑 → ran (𝐹𝑓 + 𝐺) = {𝑧 ∣ ∃𝑎𝐴 𝑧 = ((𝐹𝑎) + (𝐺𝑎))})
28 ofrn.3 . . . . 5 (𝜑+ :(𝐵 × 𝐵)⟶𝐶)
29 ffn 6198 . . . . 5 ( + :(𝐵 × 𝐵)⟶𝐶+ Fn (𝐵 × 𝐵))
3028, 29syl 17 . . . 4 (𝜑+ Fn (𝐵 × 𝐵))
31 frn 6206 . . . . . 6 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
321, 31syl 17 . . . . 5 (𝜑 → ran 𝐹𝐵)
33 frn 6206 . . . . . 6 (𝐺:𝐴𝐵 → ran 𝐺𝐵)
348, 33syl 17 . . . . 5 (𝜑 → ran 𝐺𝐵)
35 xpss12 5273 . . . . 5 ((ran 𝐹𝐵 ∧ ran 𝐺𝐵) → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
3632, 34, 35syl2anc 696 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵))
37 ovelimab 6969 . . . 4 (( + Fn (𝐵 × 𝐵) ∧ (ran 𝐹 × ran 𝐺) ⊆ (𝐵 × 𝐵)) → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3830, 36, 37syl2anc 696 . . 3 (𝜑 → (𝑧 ∈ ( + “ (ran 𝐹 × ran 𝐺)) ↔ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)))
3938abbi2dv 2872 . 2 (𝜑 → ( + “ (ran 𝐹 × ran 𝐺)) = {𝑧 ∣ ∃𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 𝑧 = (𝑥 + 𝑦)})
4018, 27, 393sstr4d 3781 1 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  {cab 2738  wrex 3043  wss 3707  cmpt 4873   × cxp 5256  ran crn 5259  cima 5261   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  𝑓 cof 7052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-of 7054
This theorem is referenced by:  sibfof  30703
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